Pitch point action gearing

ABSTRACT

A system of helical or spiral bevel gearing that by violating the &#34;law of gearing&#34; restricts the tooth action entirely to the pitch point. The greatly reduced sliding velocity between the teeth of mating gears permits them to be made of &#34;dry bearing&#34; or other moldable materials without exceeding rated PV values, and a multiplicity of contact points moving transversely along the pitch line insures continuity of action.

This invention relates to the profile and shape of gear teeth. Thegearing system herein disclosed in intended to be especially adapted togears molded from dry bearing materials, but its advantages are suchthat it will find many applications in gears made of a wide variety ofmaterials, including elastomers, plastics, die casting alloys, sinteredmetals and hardened steel, as well as combinations of these materials.

Designing gears of reasonable power capacity and wear life which can bemade of moldable materials has generally been considered animpossibility. The problem has been that the involute system of gearing,because it has proven to be most satisfactory for cut gearing, mustnecessarily be the optimum system for molded gearing. This is anincorrect assumption, and the present invention is intended to disclosea new system of gearing that is considerably more appropriate for moldedgearing than the involute system.

As a class, moldable materials have certain properties which differ fromthose of the harder materials from which gears are customarily machined(steel, cast iron, bronze). One of these properties, allowablecompressive stress, is considerably lower than that for the hardmaterials, and it has generally been assumed that because of this thecapacity of molded gearing could never be very great. This assumption isalso incorrect, because moldable materials are generally softer (have alower modulus of elasticity) than cut gear materials, so that toothsurface loads tend to be distributed over much larger areas. Inaddition, local deformations are much greater, so that considerablywider gear faces may be used without the inequalities of loading whichoccur in hard gears as a result of slight axial misalignment. Betweenthe larger surface areas of contact and the wider gear faces possiblewith moldable materials, gears can readily be designed that willaccommodate tooth loads substantially as great as those of the best heattreated steel.

The real problem of making satisfactory molded gearing is, therefore,not one of limited tooth load, but is rather a problem of low resistanceto abrasion and low thermal conductivity. These problems have long beenrecognized in the design of molded bearings, where they have been takeninto consideration by means of what is called the PV factor. Thisquantity, which is the product of bearing pressure in pounds per squareinch and sliding velocity in feet per minute, reflects both the rate ofwear in a rubbing contact and, in extreme service, the power convertedto frictional heat per unit projected area of bearing surface which mustbe dissipated if the bearing is not to be destroyed by overheating.

PV factors for molded bearings vary considerably, from a few hundred toas high as thirty or forty thousand, depending on the materials,lubrication, ambient temperatures, lining thickness, required wear life,etc. A typical value, for molybdenum disulfied filled nylon,unlubricated, would be about 8,000.

The example of a dry bearing material with a PV of 8,000 is instructiveto consider in connection with the unsuitability of the involute systemfor molded gearing. This particular material has a compressive strengthof 20,000 p.s.i., and if this is to be fully utilized, and yet theallowable PV is not to be exceeded, the sliding velocity for the teethshould not exceed 8,000/20,000 or 0.4 feet per minute.

If involute gears are designed for minimum surface stress in the toothcontact area, they must be designed with the minimum number of teeththat can be mated without tooth interference. This means that engagementstarts substantially at the point where the line of action is tangent tothe base circle of the driving gear and continues to the point where theaddendum circle of the driving gear intersects the line of action. Thisproduces an angle of action for 20° stub teeth of about 36°, or 1/10revolution, and a sliding velocity V_(s) between mating teeth whichstarts at ##EQU1## where V is the pitch line velocity, R₁ and R₂ thepitch radii of the gears, and φ the pressure angle. From this maximumvalue at the point of engagement, which amounts to about two thirds ofthe pitch line velocity if the gears have equal diameters and 20° teeth,the sliding velocity reduces almost linearly to zero at the pitch point,then increases to almost the same as the starting value at the end ofthe contact angle. Hence the most serious condition is for the flanksand tips of the mating teeth, and this is the condition which limits theutility of the gears.

With regard to actual values of sliding velocity, it is probablyreasonable to consider an average service for power transmission gearingto be perhaps a 6-inch gear attached to a 1725 r.p.m. induction motor(or a 3-inch gear attached to a 3450 r.p.m. motor). This gives a pitchline velocity of about 2700 feet per minute, and a maximum slidingvelocity at the initiation of tooth contact of about 1800 feet perminute. When this value is compared to the 0.4 foot per minute thatwould be recommended on the basis of the PV for molded nylon gears, itbecomes evident why such gears have been of necessity restricted to"light service." If the sliding velocity of 1800 feet per minute is tobe taken into consideration, the compressive load on the teeth should bereduced to 8000/1800 or about 4.4 p.s.i., instead of 20,000 p.s.i.

Some modifications in the above evaluation should be made to takeaccount of the fact that the critical combination of load and slidingvelocity is only imposed on a particular point on a tooth for a fewdegrees of each 360° revolution. This would lead to longer service livesthan would be obtained from continuously operated bearings of equivalentPV loading, or, alternatively somewhat higher PV values could beemployed in gear design to give corresponding service lives. Howeverthese corrections are nowhere near sufficient to make up for the factthat for rotational speeds normally encountered in most powertransmission applications, involute tooth shapes impose too high slidingvelocities to permit the use of molded gearing in any circumstancesexcept those which impose tooth loads that are only a fraction of thetooth surface compressive strength.

The object of the present invention is therefore to provide a new toothform that will cause the sliding velocity between mating teeth to be assmall as possible negative. This would simply mean that the toothprofile of negative radius of curvature was concave instead of convex,and had its center of curvature between T₁ and Q₁, or T₂ and Q₂. Unlikethe Wildhaber-Novikov system, however (U.S. Patent No. 1,601,750;Product Engineering, Sept. 17, 1962, p. 91), there is no specialadvantage in mating concave and convex teeth. In the gearing hereindisclosed, the maximum relative radius of curvature r that can be usedwithout interference and with rapid profile separation on either side ofthe pitch point P is the same for both convex-concave and convex-convexteeth, namely the value indicated in Equation 2. Since the Hertziancontact stress is based on relative radius of curvature r, it followsthat the only effect of using concave-convex profiles in the presentinvention is a loss in interchangeability without any compensating gainin allowable tooth load.

With regard to the Wildhaber-Novikov system, it may be noted that whileit allows an increase in tooth load relative to the involute system inthe case of lubricated cut gearing, it is basically even less suitablefor unlubricated molded gearing than the involute system. This isbecause the Wildhaber-Novikov system places the centers of curvature ofthe mating concave-convex profiles substantially at the pitch point, sothat the contact surfaces must be considerably removed from the pitchpoint and consequently lie in a region of relatively rapid sliding. Theaverage sliding velocity for a typical Wildhaber-Novikov gear is abouthalf the pitch line velocity, as compared to one third for the involutesystem.

In contrast to these average sliding velocities of thirty to fiftypercent of the pitch line velocity for typical involute andWildhaber-Novikov gears respectively, the average sliding velocity forgears embodying the herein disclosed invention is of the order of onepercent of the pitch line velocity, and the increase in capacity basedon PV values for molded dry bearing materials will be commensurate. Thatthere is any sliding velocity at all in the subject gearing system isdue to the fact that the moduli of elasticity of moldable materials issuch that the tooth contact extends over a relatively large area, onlythe center of which has zero velocity by virtue of its being exactly atthe pitch point. The edges of the contact area may be a degree or so ofrotation in front of or beyond the pitch point, depending on themagnitude of the helix or spiral angle. But these fringe areas that aresubjected to minor sliding are more lightly loaded than the center ofthe contact area, in any case.

It should also be noted, in connection with Equations 2 and 3, that R₁or R₂ may go to infinity in the case of a rack, and r₁ or r₂ may also goto infinity if one of any pair of mating gears has flat teeth; theequations are still valid for these conditions, however.

With regard to the other drawings, FIG. 2 is a plan view of gear 1 ofFIG. 1, showing a herringbone construction. It will be noted that theface length is quite wide relative to the gear diameter, since there areno serious alignment problems in this system. Also, since unit costs arenot greatly increased by using herringbone teeth, and the face is amplywide, advantage may be taken of the herringbone construction toeliminate external axial thrust.

FIG. 3 shows the front and back tooth profiles in a sectional plane ofFIG. 2 normal to the tooth directions. As in conventional helicalgearing, the pressure angle φ_(n) is related to that in the plane ofrotation by the standard equation ##EQU2## where ψ is the helix angle.Similarly, the radius of curvature in the normal plane r_(n1) relates tothat in the plane of rotation r₁ as follows: ##EQU3## For a pair ofmating gears, the relative radius of curvature r_(n) will be: ##EQU4##where r is the relative radius of curvature in the plane of rotation asgiven by Equation 3. In practice, the most useful pressure and helixangles are such (20 to 40°) that the argument of Equation 6 is not farfrom unity, and since the teeth in this system tend to be quite fine,profiles cut to an exact circular arc in the normal plane will besubstantially circular arcs in the plane of rotation, and vice versa.

FIG. 4, showing an enlarged section of a tooth 3 in the normal plane,indicates the pressure distribution over the slant height of the tooth.Unlike the involute and epicyclic gear systems in which mating teethslide over one another, the teeth in this system are subjected to almostno sliding and hence the tooth load should be distributed oversubstantially the full slant height of the tooth. Except for teethinvolving elastomer materials, the height of the contact area 2b isusually less than 1/16 of an inch, and the circular pitchcorrespondingly small. For example, a gear such as that shown in FIG. 2made of nylon filled with molybdenum disulfide would have the followingspecifications:

    Pitch diameter            3"                                                  Length                    3"                                                  Pressure angle            30                                                  Helix angle               32°30°                                Diametral pitch           60                                                  Number of teeth           180                                                 Number of contact points across full face of                                  gear                      36                                                  Maximum surface stress    20,000 p.s.i.                                       Power capacity at 3450 r.p.m. and PV of                                       8000                      Approx. 160 HP                                      Separation rate factor    0.5                                             

The especially unusual features of this type of gear may be noted: thelarge number of teeth (180), the large number of separate contact areasin the three inch length of face (36), and the exceptionally high powercapacity for unlubricated molded plastic gearing (160 HP).

It should also be noted in connection with FIG. 4 that an accuratedetermination of the minor semiaxis b of the contact ellipse may becalculated from the general Hertz equations for pressure between bodieswith curvature in two directions (c.f., Theory of Elasticity, S.Timoshenko, McGraw-Hill, 1934, pp. 344-7). In these calculations thelengthwise radii of curvature r_(L) of the teeth in the normal sectionmay be obtained from the equation: ##EQU5## Approximate values of thesemiaxis b, however, may be obtained from the expression: ##EQU6## whereS is maximum allowable surface stress, r_(n) is from Equations 2, 3, and6, and E₁ and E₂ are the moduli of elasticity for the mating gears.

Equation 8 is useful primarily for selecting the pitch for a particularset of mating gears. Optimum capacity in this system is obtained not byusing the minimum number of teeth that can be mated withoutinterference, as in the involute system, but by using the maximum numberof teeth that can be employed without the minor axis of the contact areaellipse (2b) exceeding the slant height of the teeth. This insures thatthe teeth will have the minimum possible circular pitch, which in turnmaximizes the number of contact points for any given face width andhelix angle. This general objective of minimizing the circular pitch toobtain a large number of contact points will also be served by makingthe working profile extend over more or less equal arcs on either sideof the pitch point, as shown in FIG. 4. This further optimizes capacityby locating the maximum surface stress S at the point where the slidingvelocity is zero.

While the tooth system herein disclosed may be used for cut gearing inapplications requiring higher efficiencies than are obtainable withinvolute gearing (i.e., gears inclined to overheat due to briefintensive use, as in racing vehicles), the primary objective of theinvention is to devise a type of gear adapted to be molded. In thiscontext, as well as in the claims, a "moldable material" is intended toencompass materials capable of being formed against a finished die ormold surface with no separate finishing operation involving scraping,cutting or grinding of material from the tooth surface, as for exampleby molding, plastic casting, extruding or die casting. A "dry bearingmaterial" is one intended to be used in unlubricated journal bearings.These usually carry a PV rating and at present include such materials astetrafluoroethylene or graphite; nylon, tetrafluoroethylene or phenolicfilled with molybdenum disulfide, glass fiber, lead, or graphite;sintered materials such as bronze, aluminum, iron, or nylon filled withoil, lead tetrafluoroethylene, molybdenum disulfide or graphite orcombinations thereof; and hard materials such as steel or die castaluminum which have a low friction surface of vapor deposited dry filmor baked tetrafluoroethylene.

From the above explanation it will be evident that numerous advantagesaccrue from the gearing herein disclosed. The use of tooth profilesspecifically designed to violate the "law of gearing" enables the actionof the gearing to be entirely confined to the region of the pitch lineso that sliding velocities between mating teeth are substantially zero.The effect of this is to reduce the frictional losses to the point wheredry bearing materials may be used for the teeth as well as a variety ofsoft moldable materials of low thermal conductivity. Gears made of thesematerials will be exceptionally economical.

In addition to being economical, gears embodying the present inventionare highly efficient, extremely durable, quiet and capable oftransmitting, at least at speeds of 1725 r.p.m. and above, substantiallyas much power as the finest cut gearing. In some high speed applicationswhere the operating conditions are closely controlled, gears may bedesigned embodying the invention in such a way that their radial growthor cold flow due to centrifugal effect will substantially offset toothwear.

Further advantages of the gears embodying the system herein disclosedare that they may be made interchangeable with no loss of capacity andare relatively insensitive to axial misalignment, molding distortion orchanges in center distance.

The specific description given above of the preferred form of theinvention should not be taken as restrictive as it will be apparent thatvarious modifications in design may be resorted to by those skilled inthe art without departing from the scope of the following claims.

I claim:
 1. In a pair of mating gears,teeth formed to extend across therims of said gears in a direction slantingly disposed with respect tothe pitch line of said pair to produce at least one area of contactbetween said pair for all rotational positions of said pair, .[.saidteeth being formed so that the centroid of said area of contact liessubstantially on said pitch line for all rotational positions of saidgears..]..Iadd. said teeth having in sections perpendicular to saidpitch line active profiles of relative radius of curvature at the pitchlines less than the product of the pitch radii of said pair times thesine of the pressure angle divided by the sum of said pitch radii,addendum height of the teeth of one of said pair being greater than 0.5divided by the normal diametral pitch and the pitch and pressure angleof the teeth are such that when the maximum allowable torque is appliedto said gears each said area of contact occupies substantially the fullheight of the tooth face..Iaddend. .[.2. In a pair of mating gears,teeth formed to extent across the rims of said gears at a sufficientangle with respect to the pitch line of said pair to produce at leasttwo separate areas of contact between said pair, said teeth having insections perpendicular to said pitch line active profiles of relativeradius of curvature at the pitch line less than the product of the pitchradii of said pair times the sine of the pressure angle divided by thesum of said pitch radii..]..[.3. A pair of mating gears according toclaim 1 or 2 wherein the pitch and pressure angle of the teeth are suchthat when the maximum allowable torque is applied to said gears eachsaid area of contact occupies substantially the full height of the toothface..].
 4. A pair of mating gears according to claim 1 .[.or 2.].wherein the centers of curvature of said teeth at said area of contactare substantially removed from said pitch line.
 5. A pair of matinggears according to claim 1 .[.or 2.]. wherein the teeth of at least onegear are formed of moldable material.
 6. A pair of mating gearsaccording to claim 1 .[.or 2.]. wherein the active profiles of the teethin the normal plane are circular arcs.
 7. A pair of mating gearsaccording to claim 1 .[.or 2.]. wherein the active profiles of the teethin planes normal to the pitch line are circular arcs.
 8. A pair ofmating gears according to claim 1 .[.or 2 wherin.]. .Iadd.wherein.Iaddend.the active faces of the teeth of at least one gear are of drybearing material.
 9. A pair of mating gears according to claim 1,wherein said teeth have in sections perpendicular to the elements of therespective pitch surfaces of said gears active profiles which areintersected by said pitch surfaces at the central portions of saidprofiles .[., the relative radius of curvature of engaging segments ofsaid central portions being less than the relative radius of curvatureof conjugate profiles.].. .[.10. The method of reducing the averagesliding velocity of the teeth of mating gears below the average slidingvelocity attainable with conjugate gears of the same pitch linevelocity, comprising forming said teeth:(a) to extend across the rims ofsaid mating gears in a direction slantingly disposed with respect to thepitch line of said mating gears, (b) with active profiles which areintersected by the respective pitch surfaces of said mating gears, and(c) with active profiles of relative radius of curvature less than therelative radius of curvature of conjugate profiles..].